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Math Help - Power Series and radius/interval of convergence

  1. #1
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    Power Series and radius/interval of convergence

    Hi

    The problem is to find the radius and interval of convergence of the series...

    ((-1)^(k+1)*(x+1)^k)/k from k=1 to infinity.

    I have attempted using the ratio test for absolute convergence and have simplified it down to

    rho=lim(k->infinity) |(x+1)|
    I know it will converge absolutely if rho is less than 1 so i got |(x+1)|<1

    So is the radius of convergence is 1 and the interval of convergence -2<x<0?
    Last edited by Sam1111; October 8th 2009 at 10:15 PM.
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  2. #2
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    Quote Originally Posted by Sam1111 View Post
    Hi

    The problem is to find the radius and interval of convergence of the series...

    ((-1)^(k+1)*(x+1)^k)/k from k=1 to infinity.

    I have attempted using the ratio test for absolute convergence and have simplified it down to

    rho=lim(k->infinity) |(x+1)|
    I know it will converge absolutely if rho is less than 1 so i got |(x+1)|<1

    So is the radius of convergence is 1 and the interval of convergence -2<x<0?
    Almost: the interval of convergence contains both 0 and -2.

    Tonio
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  3. #3
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    Ok thanks! I was just making sure I was on the right track before checking the end points
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  4. #4
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    Quote Originally Posted by tonio View Post
    Almost: the interval of convergence contains both 0 and -2.

    Tonio
    That's not correct. If x= -2, the series is \sum_{k=1}^\infty \frac{(-1)^{k+1}(-1)^k}{k}= -\sum_{k=1}^\infty \frac{1}{k} which does not converge. The series converges for -2< x\le 0.
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